pwm {lmomco} | R Documentation

## Unbiased Sample Probability-Weighted Moments

### Description

Unbiased sample Probability-Weighted Moments (PWMs) are computed from a sample. The first five β_r's are computed by default.

β_r = n^{-1}sum^n_{j=1} (j-1 choose r)x_{j:n}

### Usage

    pwm(x,nmom=5,sort=TRUE)

### Arguments

`x` | A vector of data values.
`nmom` | Number of PWMs to return.
`sort` | Does the data need sorting? The computations require sorted data. This option is provided to optimize processing speed if presorted data already exists.

### Value

An R `list` is returned.

`betas` | The PWMs. Note that convention is the have a β_0, but this is placed in the first index `i=1` of the `betas` vector.
`source` | Source of the PWMs: “pwm”

### Author(s)

W.H. Asquith

### References

Greenwood, J.A., Landwehr, J.M., Matalas, N.C., and Wallis, J.R., 1979, Probability weighted moments—Definition and relation to parameters of several distributions expressable in inverse form: Water Resources Research, vol. 15, p. 1,049–1,054.

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, vol. 52, p. 105–124.

Hosking, J.R.M. and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

### See Also

`lmoms`, `pwm2lmom`, `pwm`

### Examples

    # Data listed in Hosking (1995, table 29.2, p. 551)
    H <- c(3,4,5,6,6,7,8,8,9,9,9,10,10,11,11,11,13,13,13,13,13,
           17,19,19,25,29,33,42,42,51.9999,52,52,52)
    # 51.9999 was really 52, but a real non censored data point.
    z <-  pwmRC(H,52,checkbetas=TRUE)
    str(z)
    # Hosking(1995) reports that A-type L-moments for this sample are
    # lamA1=15.7 and lamAL-CV=.389, and lamAL-skew=.393
    pwm2lmom(z$Abetas)
    # My version of R reports 15.666, 0.3959, and 0.4030

    # See p. 553 of Hosking (1995)
    # Data listed in Hosking (1995, table 29.3, p. 553)
    D <- c(-2.982, -2.849, -2.546, -2.350, -1.983, -1.492, -1.443,
           -1.394, -1.386, -1.269, -1.195, -1.174, -0.854, -0.620,
           -0.576, -0.548, -0.247, -0.195, -0.056, -0.013,  0.006,
            0.033,  0.037,  0.046,  0.084,  0.221, 0.245, 0.296)
    D <- c(D,rep(.2960001,40-28)) # 28 values, but Hosking mentions 40 values in total
    z <-  pwmRC(D,.2960001)
    # Hosking reports B-type L-moments for this sample are
    # lamB1 = -.516 and lamB2 = 0.523
    pwm2lmom(z$Bbetas)
    # My version of R reports -.5162 and 0.5218

[Package lmomco version 0.92 Index]